3.115 \(\int \frac{1}{x^2 (a^2+2 a b x^3+b^2 x^6)^{5/2}} \, dx\)

Optimal. Leaf size=398 \[ -\frac{455 \left (a+b x^3\right )}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}+\frac{13}{108 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac{1}{12 a x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

455/(972*a^4*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1
3/(108*a^2*x*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 65/(324*a^3*x*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6]) - (455*(a + b*x^3))/(243*a^5*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/3)*(a + b*x^3)*ArcTan[(
a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/
3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(729*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (455*b^(1/3)*(a + b*
x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

________________________________________________________________________________________

Rubi [A]  time = 0.216311, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {1355, 290, 325, 292, 31, 634, 617, 204, 628} \[ -\frac{455 \left (a+b x^3\right )}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}+\frac{13}{108 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac{1}{12 a x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

455/(972*a^4*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1
3/(108*a^2*x*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 65/(324*a^3*x*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6]) - (455*(a + b*x^3))/(243*a^5*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/3)*(a + b*x^3)*ArcTan[(
a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/
3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(729*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (455*b^(1/3)*(a + b*
x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^3\right )^5} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{1}{12 a x \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (13 b^3 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^3\right )^4} \, dx}{12 a \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{1}{12 a x \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (65 b^2 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^3\right )^3} \, dx}{54 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{1}{12 a x \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (455 b \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^3\right )^2} \, dx}{324 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (455 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^3\right )} \, dx}{243 a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \left (a+b x^3\right )}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (455 b \left (a b+b^2 x^3\right )\right ) \int \frac{x}{a b+b^2 x^3} \, dx}{243 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \left (a+b x^3\right )}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (455 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (455 \left (a b+b^2 x^3\right )\right ) \int \frac{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \left (a+b x^3\right )}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (455 \left (a b+b^2 x^3\right )\right ) \int \frac{-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{1458 a^{16/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (455 \sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac{1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{486 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \left (a+b x^3\right )}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (455 \left (a b+b^2 x^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{243 a^{16/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \left (a+b x^3\right )}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}

Mathematica [A]  time = 0.130211, size = 242, normalized size = 0.61 \[ \frac{\left (a+b x^3\right ) \left (-910 \sqrt [3]{b} \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-1179 a^{4/3} b x^2 \left (a+b x^3\right )^2-594 a^{7/3} b x^2 \left (a+b x^3\right )-243 a^{10/3} b x^2-\frac{2916 \sqrt [3]{a} \left (a+b x^3\right )^4}{x}-2544 \sqrt [3]{a} b x^2 \left (a+b x^3\right )^3+1820 \sqrt [3]{b} \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-1820 \sqrt{3} \sqrt [3]{b} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{2916 a^{16/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

((a + b*x^3)*(-243*a^(10/3)*b*x^2 - 594*a^(7/3)*b*x^2*(a + b*x^3) - 1179*a^(4/3)*b*x^2*(a + b*x^3)^2 - 2544*a^
(1/3)*b*x^2*(a + b*x^3)^3 - (2916*a^(1/3)*(a + b*x^3)^4)/x - 1820*Sqrt[3]*b^(1/3)*(a + b*x^3)^4*ArcTan[(-a^(1/
3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] + 1820*b^(1/3)*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] - 910*b^(1/3)*(a +
b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(16/3)*((a + b*x^3)^2)^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.02, size = 536, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

-1/2916*(-1820*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^13*b^4-1820*ln(x+(a/b)^(1/3))*x^13
*b^4+910*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^13*b^4+5460*(a/b)^(1/3)*x^12*b^4-7280*3^(1/2)*arctan(1/3*3^(1/2)*
(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^10*a*b^3-7280*ln(x+(a/b)^(1/3))*x^10*a*b^3+3640*ln(x^2-(a/b)^(1/3)*x+(a/b)^(
2/3))*x^10*a*b^3+20475*(a/b)^(1/3)*x^9*a*b^3-10920*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*
x^7*a^2*b^2-10920*ln(x+(a/b)^(1/3))*x^7*a^2*b^2+5460*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^7*a^2*b^2+28080*(a/b)
^(1/3)*x^6*a^2*b^2-7280*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^4*a^3*b-7280*ln(x+(a/b)^(
1/3))*x^4*a^3*b+3640*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^4*a^3*b+16224*(a/b)^(1/3)*x^3*a^3*b-1820*3^(1/2)*arct
an(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x*a^4-1820*ln(x+(a/b)^(1/3))*x*a^4+910*ln(x^2-(a/b)^(1/3)*x+(a/
b)^(2/3))*x*a^4+2916*(a/b)^(1/3)*a^4)*(b*x^3+a)/(a/b)^(1/3)/x/a^5/((b*x^3+a)^2)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5945, size = 724, normalized size = 1.82 \begin{align*} -\frac{5460 \, b^{4} x^{12} + 20475 \, a b^{3} x^{9} + 28080 \, a^{2} b^{2} x^{6} + 16224 \, a^{3} b x^{3} + 2916 \, a^{4} + 1820 \, \sqrt{3}{\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + 910 \,{\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 1820 \,{\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right )}{2916 \,{\left (a^{5} b^{4} x^{13} + 4 \, a^{6} b^{3} x^{10} + 6 \, a^{7} b^{2} x^{7} + 4 \, a^{8} b x^{4} + a^{9} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/2916*(5460*b^4*x^12 + 20475*a*b^3*x^9 + 28080*a^2*b^2*x^6 + 16224*a^3*b*x^3 + 2916*a^4 + 1820*sqrt(3)*(b^4*
x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4*x)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b/a)^(1/3) - 1/3*
sqrt(3)) + 910*(b^4*x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4*x)*(b/a)^(1/3)*log(b*x^2 - a*x*(b/
a)^(2/3) + a*(b/a)^(1/3)) - 1820*(b^4*x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4*x)*(b/a)^(1/3)*l
og(b*x + a*(b/a)^(2/3)))/(a^5*b^4*x^13 + 4*a^6*b^3*x^10 + 6*a^7*b^2*x^7 + 4*a^8*b*x^4 + a^9*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(1/(x**2*((a + b*x**3)**2)**(5/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x